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arXiv:0901.3155 · American Journal of Physics

Particle in a box with a δ-function potential: Strong and weak coupling limits

Yogesh N. Joglekar

Published Jul 13, 2009• 11 citations
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Abstract

A particle in a box with the δ-function potential Vλ(x)=λδ(x−x0) has been recently explored. We show that this example is solvable in the weak (λ→0±) and the strong (1∕λ→0±) coupling limits. In either limit the attractive and repulsive potentials lead to identical spectra, with the possible exception of a single negative-energy state that is present when 1∕λ→0−. We numerically obtain the spectra near the strong-coupling limit and discuss the consequences of the degeneracy that arises when 1∕λ→0±.

DOI
10.1119/1.3119178

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A particle in a box with the δ-function potential Vλ(x)=λδ(x−x0) has been recently explored. We show that this example is solvable in the weak (λ→0±) and the strong (1∕λ→0±) coupling limits. In either limit the attractive and repulsive potentials lead to identical spectra, with the possible exception of a single negative-energy state that is present when 1∕λ→0−. We numerically obtain the spectra near the strong-coupling limit and discuss the consequences of the degeneracy that arises when 1∕λ→0±.
The Schrödinger equation for a particle in the V-shaped potential decorated by a repulsive or attractive Dirac delta function interaction at the center is solved, demonstrating the crucial influence of point interaction on the even-parity states of the original system without decoration. As strength of the attraction increases, the ground state energy falls down without limit; and in limit of infinitely large attraction, the ground state approaches a singular state. Our analysis and conclusion can be readily generalized to any one-dimensional system a particle interacts with symmetrical potential plus the Dirac delta function interaction at the center.
The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.
The Fredholm equations for one-dimensional two-component fermions with repulsive and with attractive $\ensuremath{\delta}$-function interactions are solved by an asymptotic expansion for strong repulsion, weak repulsion, weak attraction, and strong attraction. Consequently, we obtain the first few terms of the expansion of the ground-state energy for the Fermi gas with polarization for these regimes. We also prove that the two sets of Fredhom equations for weakly repulsive and attractive interactions are identical as long as the integration boundaries match each other between the two types. Thus the asymptotic expansions of the energies of repulsive and attractive fermions are identical to all orders in this region. The identity of the asymptotic expansions may not mean that the energy connects analytically.